1. Field of the Invention
The present invention relates to a method for quantifying uncertainties related to continuous and discrete parameters descriptive of a medium such as an underground zone and/or for managing the selection of a scenario from a series of possible scenarios relative to this medium, by construction of experiment designs and results analysis suited to the experiment designs constructed.
2. Description of the Prior Art
The documents mentioned hereafter are representative of the prior art:    Benoist D., Tourbier Y. and Germain-Tourbier S. (1994). Plans d'Expériences Construction et Analyse. Technique & Documentation-Lavoisier, Paris;    Cox D. R. (1984). Present Position and Potential Developments: Some Personal Views: Design of Experiments and Regression. J. of Royal Statistical Society, Ser. A, 147, pp. 306-315;    Dejean J.-P., Blanc G. (1999). Managing Uncertainties on Production Predictions Using Integrated Statistical Methods. SPE 56696, SPE Annual Technical Conference and Exhibition, Houston, 3-6 Oct. 1999;    Draper N. R. and John J. A. (1988). Response-Surface Designs for Quantitative and Qualitative Variables. Technometrics, 30(4), pp. 423-428;    Droesbeke J.-J., Fine J. and Saporta G. (1997). Plans d'Expériences: Applications à l'Entreprise. Technip, Paris    Montgomery D. C. and Peck E. A. (1992). Introduction to Linear Regression Analysis. Wiley Series in Probability and Mathematical Statistics, New York;    Wu C. F. J. and Ding Y. (1998). Construction of Response Surface Designs for Qualitative and Quantitative Factors. J. of Planning and Inferences, 71, pp. 331-348;    Zabalza I., Dejean J.-P., Collombier D. (1998). Prediction and Density Estimation of a Horizontal Well Productivity Index Using Generalizes Linear Model. ECMOR VI, Peebles, 8-11 Sep. 1998;    Zabalza-Mezghani I. (2000). Analyse statistique et planification d'expérience en ingenierie de reservoir. These de doctorant de 3ème cycle, Université de Pau.
The methods for organizing experiment designs generally aim to best plan the experiments or tests to be carried out so as to establish relations between various causes or factors (here the permeability, the porosity, interactions, etc.) and the responses studied (here the cumulative volume of oil, the water cut, etc.) and to derive, if possible, predictive models. In the description hereafter, the analytical (conventionally polynomial) model resulting from the adjustment of experimental results is referred to as “response surface”. These methods generally comprise the construction of experiment designs that have to be performed to establish these relations and an analysis of the results.
Various studies have been carried out by Dejean J.-P. et al. (1999), Zabalza I. (1998) and Zabalza-Mezghani I. (2000) in order to quantify the uncertainties on physical parameters of underground hydrocarbon reservoirs such as the porosity, the permeability, the position of the well, the drilled well length, the structure of the heterogeneities by geostatistical modelling, etc., which use the experiment design method and statistical methods.
Although the method resulting from these studies allows dealing with the continuous physical parameters (quantitative factors), it allows taking into account discrete parameters (qualitative factors) such as the status of a fault in the reservoir for example only by repeating the same experiment design as many times as there are scenarios to be compared. The simulation cost then quickly becomes prohibitive. Furthermore, from the studies being carried out separately on each scenario, it is impossible to take into account the effect due to the discrete parameters and thus to quantify the uncertainty related to the scenarios. In the text hereafter, the possible states of a discrete parameter are referred to as “modalities”. The scenarios result from the combination of the modalities of discrete parameters. For example, a discrete parameter with two modalities and a discrete parameter with three modalities generate six scenarios. The engineer can choose a scenario if the discrete parameters are controllable (completion levels, etc.) or have no action on the scenarios if a discrete parameter is not controllable (status of a fault, etc.).
The construction of designs integrating both quantitative and qualitative factors has notably been dealt with by Cox (1984). The objectives to be fulfilled by these designs are defined by Draper N. R. et al. (1988). The experiment design construction method defined by Wu C. F. J. et al. (1998) proposes for example fixing for the simulation the quantitative factor levels by means of a composite design (well-known in the art and described in any manual on experiment designs) to which columns representing the qualitative factors are added. These columns are determined from quantitative criteria (optimization of conventional experiment design criteria, D-optimality). In practice, the authors have constructed designs allowing integration of a single discrete parameter with two modalities. Their construction method, based on the numerical optimization of a quality criterion, rapidly reaches its limits when the number of scenarios increases. In fact, the algorithmic cost of determination of such a design cannot be considered in practice.
In the prior art, the results obtained by applying experiment designs are analyzed according to a conventional scheme notably described by Benoist D. et al. (1994) or by Droesbeke J.-J. et al. (1997). This method is perfectly suited to the quantitative factors but it does not allow fully dealing with experiment designs involving quantitative and qualitative factors. There is in fact a loss of information which should be exploited, notably within the context of uncertainties quantification in reservoir engineering.
Conventional analysis of experiment design results cannot be applied in this context, for two main reasons. First, the designs proposed allow adjustment of not only one but several distinct models (global model including the various scenarios and marginal models for each scenario). The model(s) that will provide the most information during the result analysis stages therefore have to be determined. On the other hand, it is necessary to fully understand the role of the discrete parameters in the global model (simple effect and interactions of the qualitative factors, repercussion on the response).
In the context defined above, the main quality required for economical experiment designs integrating both quantitative and qualitative factors is to have the necessary properties for good adjustment of the response surface to the quantitative factors. The structure of the conventional composite design is suited to this type of problem. However, it cannot be applied as it is when there are qualitative factors. In fact, it requires fixing five levels (arranged on a ratio scale) for each factor, which is not possible with the modalities of the qualitative factors (for example two modalities for the status of a fault, open or closed).